TB Cluster Models, Time Scales and Relations to HIV

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TB Cluster Models, Time Scales and Relations to
HIV
Carlos Castillo-Chavez
Department of Biological Statistics and
Computational Biology Department of Theoretical
and Applied Mechanics Cornell University, Ithaca,
New York, 14853
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Outline

• A non-autonomous model that incorporates the
  impact of HIV on TB dynamics.
• Model to test CDCs TB control goals.
• Casual versus close contacts and their impact on
  TB.
• Time scales and singular perturbation approaches
  in the study of the dynamics of TB.

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TB in the US(1953-1999)
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Reemergence of TB

• New York City and San Francisco had
• recent outbreaks.
• Cost of control the outbreak in NYC alone
• was estimated to be about 1 billion.
• Observed national TB case rate increase.
• TB reemergence became an international
• issue.
• CDC sets control goal in 1989.

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Basic Model Framework

• NSEIT, Total population
• F(N) Birth and immigration rate
• B(N,S,I) Transmission rate (incidence)
• B(N,S,I) Transmission rate (incidence)

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Model Equations
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TB control in the U.S.

• CDC Short-Term Goal
• 3.5 cases per 100,000 by 2000.
• Has CDC met this goal?
• CDC Long-term Goal
• One case per million by 2010.
• Is it feasible?

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Model Construction
Since d has been approximately equal to zero over
the past 50 years in the US, we only consider
Hence, N can be computed independently of TB.
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Non-autonomous model (permanent latent class of
TB introduced)
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Effect of HIV
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Upper Bound and Lower Bound For Epidemic
Threshold
If R?lt1, L1(t), L2(t) and I(t) approach zero If
R?gt1, L1(t), L2(t) and I(t) all have lower
positive boundary If ?(t) and d(t) are
time-independent, R? and R? are Equal to R0 .
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Parameter estimation and simulation setup
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Parameter estimation and simulation setup
N(t) is from census data and population projection
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RESULTS
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CONCLUSIONS
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CONCLUSIONS
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CDCs Goal Delayed

• Impact of HIV.
• Lower curve does not include HIV impact
• Upper curve represents the case rate when HIV is
  included
• Both are the same before 1983. Dots represent
  real data.

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Regression approach
A Markov chain model supports the same result
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Cluster Models

• Incorporates contact type (close vs. casual) and
  focus on the impact of close and prolonged
  contacts.
• Generalized households become the basic
  epidemiological unit rather than individuals.
• Use natural epidemiological time-scales in model
  development and analysis.

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Close and Casual contacts
Close and prolonged contacts are likely to be
responsible for the generation of most new cases
of secondary TB infections. A high school
teacher who also worked at library infected the
students in her classroom but not those who came
to the library. Casual contacts also lead to
new cases of active TB. WHO gave a warning in
1999 regarding air travel. It announced that
flights of more than 8 hours pose a risk for TB
transmission.
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Transmission Diagram
                         
 
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Key Features

• Basic epidemiological unit cluster (generalized
  household)
• Movement of kE2 to I class brings nkE2 to N1
  population, where by assumptions nkE2(S2 /N2) go
  to S1 and nkE2(E2/N2) go to E1
• Conversely, recovery of ??I infectious bring n?I
  back to N2 population, where n?I (S1 /N1) ?? S1
  go to S2 and n?I (E1 /N1) ?? E1 go to E2

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Basic Cluster Model
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Basic Reproductive Number
where
is the expected number of infections produced by
one infectious individual within his/her
cluster. denotes the fraction who survives the
latency period and become active cases.
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Diagram of Extended Cluster Model
                           

 
 
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? (n)
Both close casual contacts are included in the
extended model. The risk of infection per
susceptible, ? , is assumed to be a nonlinear
function of the average cluster size n. The
constant p measures the average proportion of
the time that an individual spends in a cluster.
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R0 Depends on n in a non-linear fashion
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Role of Cluster Size
E(n) denotes the ratio of within cluster to
between cluster transmission. E(n) increases and
reaches its maximum value at The cluster size
n is defined as optimal as it maximizes the
relative impact of within to between cluster
transmission.
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Hoppensteadts Theorem(1973)
Reduced system
where x ? Rm, y ? Rn and ? is a positive real
parameter near zero (small parameter). Five
conditions must be satisfied (not listed here) to
apply the theorem. In addition, it is shown that
if the reduced system has a globally
asymptotically stable equilibrium then the full
system has a g.a.s. equilibrium whenever 0lt ?
ltlt1.
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Two time Scales

• Latent period is long and roughly has the same
  order of magnitude as that associated with the
  life span of the host.
• Average infectious period is about six months
  (wherever there is treatment, is even shorter).

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Rescaling
Time is measured in average infectious periods
(fast time scale), that is, ? k t. The state
variables are rescaled as follow
Where ? ? ?/? is the asymptotic carrying capacity.
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Rescaled Model
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Rescaled Model
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Dynamics on Slow Manifold

• Solving for the quasi-steady states y1, y2 and y3
  in
• terms of x1 and x2 gives
• Substituting these expressions into the equations
  for
• x1 and x2 lead to the equations of motion on the
  slow
• manifold.

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Slow Manifold Dynamics

• Where is the number of secondary
• infections produced by one infectious individual
  in a
• population where everyone is susceptible

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Theorem
If Rc0 ? 1,the disease-free equilibrium (1,0) is
globally asymptotically stable. While if Rc0 gt 1,
(1,0) is unstable and the endemic equilibrium
is globally asymptotically stable. This
theorem characterizes the dynamics on the slow
manifold
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Dynamics for Full System
Theorem For the full system, disease-free
equilibrium is globally asymptotically stable
whenever R0c lt1 while R0c gt1 there exists a
unique endemic equilibrium which is globally
asymptotically stable.
Proof approach Construct Lyapunov function for
the case R0c lt1 for the case R0c gt1, we use
Hoppensteadts Theorem. A similar result can be
found in Z. Fengs 1994, Ph.D. dissertation.
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Bifurcation Diagram

• Global bifurcation diagram when 0lt?ltlt1 where ?
  denotes
• the ratio between rate of progression to active
  TB and the
• average life-span of the host (approximately).

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Numerical Simulations
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Conclusions from cluster models

• TB has slow dynamics but the change of
• epidemiological units makes it possible to
  identify
• non-traditional fast and slow dynamics.
• Quasi steady assumptions (adiabatic
  elimination of
• parameter) are valid (Hoppensteadts
  theorem).
• The impact of close and casual contacts can be
  study
• using this approach as long as progression
  rates
• from the latently to the actively-infected
  stages are
• significantly different.

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Conclusions from cluster models

• Singular perturbation theory can be used to study
  the global asymptotic dynamics.
• Optimal cluster size highlights the relative
  impact of close versus casual contacts and
  suggests alternative mechanisms of control.
• The analysis of the system for the case where the
  small parameter ? is not small has not been
  carried out. Simulations suggest a wider range.